The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915257384960000 |
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| author | Faulhuber, Markus Gumber, Anupam Shafkulovska, Irina |
| author_facet | Faulhuber, Markus Gumber, Anupam Shafkulovska, Irina |
| contents | We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic-geometric mean iterations. This follows from connections to Jacobi theta functions and Ramanujan's corresponding theories. As a consequence we re-discover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau's constant is obtained as the cubic arithmetic-geometric mean of $\sqrt[3]{2}$ and 1, which we believe to be a new result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_04202 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators Faulhuber, Markus Gumber, Anupam Shafkulovska, Irina Classical Analysis and ODEs Functional Analysis Metric Geometry Number Theory 33C05, 33C67, 33C80, 42C15 We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic-geometric mean iterations. This follows from connections to Jacobi theta functions and Ramanujan's corresponding theories. As a consequence we re-discover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau's constant is obtained as the cubic arithmetic-geometric mean of $\sqrt[3]{2}$ and 1, which we believe to be a new result. |
| title | The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators |
| topic | Classical Analysis and ODEs Functional Analysis Metric Geometry Number Theory 33C05, 33C67, 33C80, 42C15 |
| url | https://arxiv.org/abs/2209.04202 |